= Centripetal subtraction error
<Centripetal subtraction error> is the residual error left when a modeled centripetal contribution is removed from a <rotating accelerometer> signal. For the positive magnitude $C=r\omega^2$, small errors in radius and <angular speed> give
$$
\Delta C\approx\omega^2\Delta r+2r\omega\Delta\omega.
$$
The residual's sign depends on whether the correction is overestimated or underestimated. A correct model also needs synchronized speed information and the appropriate sensor-axis projection; a primary rotating-sensor experiment documents the importance of radial/tangential directions and rotating-frame effects.<Evaluation of MEMS-Based Wireless Accelerometer Sensors in Detecting Gear Tooth Faults in Helicopter Transmissions by David G. Lewicki and others>[Printed pp.19–21, measurement directions and spectral interpretation]{ref}
Using first-order <propagation of uncertainty>, the standard uncertainty satisfies
$$
u_C^2\approx\omega^4u_r^2+4r^2\omega^2u_\omega^2
+4r\omega^3\operatorname{cov}(r,\omega).
$$
The <covariance> term must be included when the estimates are correlated.<Evaluation of measurement data — Guide to the expression of uncertainty in measurement by JCGM>[§§5.1–5.2, sensitivity coefficients; §5.2.2, equations13–16, correlated inputs]{ref} For $r=0.020\,\mathrm m$, $\omega=100\,\mathrm{rad/s}$, $u_r=10\,\mathrm{\mu m}$ and $u_\omega=0.1\,\mathrm{rad/s}$, the two uncorrelated contributions have magnitudes0.10 and0.40m/s², giving $u_C\approx0.4123\,\mathrm{m/s^2}$.
A <centripetal timing error> can remain even with accurate individual channels. If the speed used for correction lags by $\Delta t$, then $\Delta\omega\approx-\dot\omega\Delta t$. At the same radius and speed, acceleration rate $\dot\omega=10\,\mathrm{rad/s^2}$ and lag5ms produce correction error approximately $-0.20\,\mathrm{m/s^2}$. Channel timing therefore belongs in the measurement model. These linearized formulas describe small errors; they do not replace accounting for other motion or <specific force> contributions.
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